Variance attribution calculation from a covariance matrix

Question

Say I have a portfolio with two assets with weights $(x, y)$, and the covariance matrix of the two asset is $((a, r)(r, b))$. Then the total portfolio variance would be $x^2a+2xyr+y^2b$. It is easy to get that the percentage of the variance due to asset $x$ is $\frac{x^2a+xyr}{x^2a+2xyr+y^2b}$. I wonder in the n-dimensions cases, how to calculate the variance percentage for each asset mathematically based on the covariance matrix?

Answer 1

Suppose the covariance matrix is $V$ (which is n by n) and the weights are $w$ (of length n).

Then the Portfolio Variance is $V_p = w^T V w$

and the Risk Contribution (in terms of variance) of asset $k$ is

$RC_k=w_k \sum_j V[k,j]w_j$

in words this is "the weight of asset k times the inner product of the k-th row of $V$ and the weight vector". (Sometimes the "inner product of ..." just mentioned is given the name the Marginal Risk Contribution of asset $k$, which leads to the compact expression $RC_k=w_k MRC_k$).

We then have the "decomposition property" that $V_p=\sum_k RC_k$ or in percentage terms

$$\sum_k \frac{RC_k}{V_p}=1$$

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If we apply this to the two by two case

$V=\begin{bmatrix} a & r \\ r & b \\ \end{bmatrix}$

and $w=\begin{bmatrix}x \\ y \end{bmatrix}$

we get that the total variance of the portfolio is $V_p=a x^2+2 r x y + b y^2$

The variance contribution of the first asset is $RC_1=x(ax+ry)$

and the percentage contribution is the ratio of these two (the latter divided by the former). This agrees with your result.

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Two good references for these results are

Edward Qian: On the Financial Interpretation of Risk Contribution: Risk Budgets Do Add Up (2005)

S. Maillard, T. Roncalli: On the properties of equally-weighted risk contributions portfolios (2009)

also often cited is

D Tasche: Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle (2008)